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Properties

Label 1-847-847.787-r0-0-0
Degree $1$
Conductor $847$
Sign $0.420 + 0.907i$
Analytic cond. $3.93345$
Root an. cond. $3.93345$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.449 + 0.893i)2-s + (−0.669 + 0.743i)3-s + (−0.595 − 0.803i)4-s + (0.532 + 0.846i)5-s + (−0.362 − 0.931i)6-s + (0.985 − 0.170i)8-s + (−0.104 − 0.994i)9-s + (−0.995 + 0.0950i)10-s + (0.995 + 0.0950i)12-s + (−0.870 − 0.491i)13-s + (−0.985 − 0.170i)15-s + (−0.290 + 0.956i)16-s + (−0.179 − 0.983i)17-s + (0.935 + 0.353i)18-s + (0.879 + 0.475i)19-s + (0.362 − 0.931i)20-s + ⋯
L(s)  = 1  + (−0.449 + 0.893i)2-s + (−0.669 + 0.743i)3-s + (−0.595 − 0.803i)4-s + (0.532 + 0.846i)5-s + (−0.362 − 0.931i)6-s + (0.985 − 0.170i)8-s + (−0.104 − 0.994i)9-s + (−0.995 + 0.0950i)10-s + (0.995 + 0.0950i)12-s + (−0.870 − 0.491i)13-s + (−0.985 − 0.170i)15-s + (−0.290 + 0.956i)16-s + (−0.179 − 0.983i)17-s + (0.935 + 0.353i)18-s + (0.879 + 0.475i)19-s + (0.362 − 0.931i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.420 + 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.420 + 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.420 + 0.907i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (787, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ 0.420 + 0.907i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6870154471 + 0.4387738887i\)
\(L(\frac12)\) \(\approx\) \(0.6870154471 + 0.4387738887i\)
\(L(1)\) \(\approx\) \(0.5935597268 + 0.3813297621i\)
\(L(1)\) \(\approx\) \(0.5935597268 + 0.3813297621i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.449 + 0.893i)T \)
3 \( 1 + (-0.669 + 0.743i)T \)
5 \( 1 + (0.532 + 0.846i)T \)
13 \( 1 + (-0.870 - 0.491i)T \)
17 \( 1 + (-0.179 - 0.983i)T \)
19 \( 1 + (0.879 + 0.475i)T \)
23 \( 1 + (-0.327 - 0.945i)T \)
29 \( 1 + (0.564 - 0.825i)T \)
31 \( 1 + (-0.749 + 0.662i)T \)
37 \( 1 + (0.953 - 0.299i)T \)
41 \( 1 + (-0.254 + 0.967i)T \)
43 \( 1 + (0.142 - 0.989i)T \)
47 \( 1 + (0.935 - 0.353i)T \)
53 \( 1 + (-0.290 - 0.956i)T \)
59 \( 1 + (0.710 - 0.703i)T \)
61 \( 1 + (0.548 - 0.836i)T \)
67 \( 1 + (0.0475 - 0.998i)T \)
71 \( 1 + (0.610 + 0.791i)T \)
73 \( 1 + (0.483 - 0.875i)T \)
79 \( 1 + (0.969 - 0.244i)T \)
83 \( 1 + (0.0855 + 0.996i)T \)
89 \( 1 + (-0.235 + 0.971i)T \)
97 \( 1 + (0.466 + 0.884i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.88542278019116858496331017909, −21.27931978218318446925782698324, −20.05594241846484226922469716831, −19.71837880067467133903400355787, −18.76859817604786320169011874313, −17.91573708643014906905215837602, −17.34549146204247527104619001879, −16.76963059316805588563670176053, −15.95199923230241339908578282976, −14.3262282106010178269600199780, −13.4686570753908119638592967338, −12.86893761663654024493534007456, −12.141355830115030760298397632141, −11.51534932766355805571656345816, −10.54340435761195433757713082170, −9.657982299040266921738141970149, −8.90280330115649613679368746919, −7.89584781475409823867256375387, −7.13603763433209200190825883976, −5.85600684503464656474569212626, −5.02295647491712544985921162067, −4.12345209003748575849492932094, −2.6074756656101026161319360902, −1.7449571158509618939637642575, −0.924084834455395485264664937456, 0.62640885570108444435430814281, 2.342950619651394574820495489393, 3.61573328543106673282618107859, 4.8636334425991467797603877782, 5.45527344435847379774170635550, 6.38254158812669957386197089484, 7.059626133568994732290410925730, 8.04455386445354305225442468700, 9.355925050265997249379236475602, 9.8296416619994300932711950966, 10.51493386199826626078785021063, 11.366032239057769227681838995701, 12.457305090287026560825422360128, 13.754527348984461613193295453028, 14.44566230672873799148635104356, 15.104417908933655834239070891264, 15.93343635195364479796465818643, 16.628109523548282528333802138031, 17.45284881182703066669997063874, 18.08369966247789395873129965053, 18.610529109994025575926877320129, 19.82828550767740536926804816494, 20.67545241470378528203748237690, 21.87593910593989561117678465238, 22.34532572483805369951099630569

Graph of the $Z$-function along the critical line